Which Textbook Best Teaches Tensor Analysis for General Relativity?

[Nusc]

Hi. I was wondering if you guys can refer me to the best possible text that teaches the computations in tensor analysis. I hate pure math. I just want to get a feel for the mathematics before engaging the physics.

As far as I know this is a widely referred text, however, it doesn't come with a solution manual.

https://www.amazon.com/gp/product/0070334846/?tag=pfamazon01-20

Thanks

 

[actionintegral]

In those Schaum books, the solutions come with the problems. But I don't learn very well by that.

I started by reading Einstein's original GR paper, just to see how much tensor stuff he actually used. Then I jumped to the last chapter of Goldstein. Field Theory by landau and lif****z came next.

I'm sure in retrospect someone else can recommend a better path!

 

[MathematicalPhysicist]

can einstein's original GR paper be viewed via the internet?

 

[Nusc]

Can someone give me a comparison between the two texts?

Tensor Analysis on Manifolds by Bishop and Goldberg, Dover Pub

Tensors and Manifolds by Wasserman.

 

[gvk]

Hi. I was wondering if you guys can refer me to the best possible text that teaches the computations in tensor analysis. I hate pure math. I just want to get a feel for the mathematics before engaging the physics...

If you have a 'physical' mind, I would recommend the book Dubrovin, Novikov, Fomenko,
Modern Geometry-Methods and Applications, Part I,II,III (Universitext)
Springer-Verlag (1990).
This is very good text with plenty of solved examples.

 

[mathwonk]

as i have recommended elsewhere, ideally one should learn coordinate free mathematics to discuss relativity. einstein did not have this available so his papers are not written in that style.

the best books in my view are those with authors who are themselves experts, which, except for the old style math, of course starts with einstein. (but actually einstein was borrowing from riemann, if you really want to start at the beginning.)

i would suggest therefore the book by thorne misner? and wheeler.

but to understand that you need to understand coordinate free calculus on m,anifolds, and befoire that coordinate free lienar algebra.

so i recommend reading an abstract linear algebra book, like maybe halmos, then an abstract calculus on manifolds book, like spivaks little one, or guillemin and pollack, which borrows from spivak, and finally the book by wheeler et al.but this is a suggestion to a bright, young, idealistic person, from an old guy who has never actually doine this himself.

better is to get a start with some lectures from a knowledgeable person. that's why school is useful. and will never be replaced by the internet.

 

[andytoh]

Can someone give me a comparison between the two texts?

Tensor Analysis on Manifolds by Bishop and Goldberg, Dover Pub

Tensors and Manifolds by Wasserman.

I have both books, and Wasserman's book is much more thorough. Here are my solutions to Chapters 1 and 2:

 

[Daverz]

I have both books, and Wasserman's book is much more thorough. Here are my solutions to Chapters 1 and 2:

Wasserman does require more abstract algebra (I have the 1st edition). And your average physics student would probably have to crack open a linear algebra text as well (a "real" math text, not one aimed at engineers.)

Given that he hates pure math, I suspect Nusc would not like Wasserman, Bishop & Goldberg, or Spivak's dense little book, but the Schaum's may be just right (well, by now he probably can tell us). Or just get a good GR text like Carroll, who is very practical about the math, from the library.

 

[Azael]

I am taking a class in general relativity now and we are using weinbergs gravitation and cosmology.
I have no other general relativity book to compare it to, but so far its one of the best textbooks I have ever had.

 

[chroot]

MTW's Gravitation is a fantastic book, and you probably won't finish it for some 18 years. Wald's General Relativity is a much more modern, but also much more demanding text.

If you need help with the mechanics of tensors, Schaum's Outline of Tensor Calculus is quite good.

- Warren

 

[Ruslan_Sharipov]

Free textbooks on-line and for downloading

Try this link http://www.freetextbooks.boom.ru" on-line and for free.

 

[Chris Hillman]

Shouldn't this thread be in the gtr board?

Hi, Nusc,

I was wondering if you guys can refer me to the best possible text that teaches the computations in tensor analysis. I hate pure math. I just want to get a feel for the mathematics before engaging the physics.

You hate pure math!? How on Earth can you hate pure math but love (I guess) theoretical physics?

About your question, I am fairly confident that most textbook authors would tell you not to worry about tensor calculus. You can pick up what you need as you go along.

Since you are apparently worried about understanding the mathematical background more than many students, though, I would probably recommend MTW. Some of the other books would also be good choices for you, I think: Weinberg is certainly a gold mine of information, as is MTW, but the easiest book for you might be Ohanian and Ruffini. Schutz makes a particular effort to introduce the idea of tensors gently, BTW, so you might also want to look at that. Carroll has some excellent material on mathematical background in some appendices which can be read semi-inpendently. All of these are excellent books, although some claim to find MTW or Weinberg daunting (I feel such judgements are mostly wrong). Most people seem to find O&R, Schutz, and Carroll to be very readable. Schutz does have partial solutions to the (excellent) problems, and you can always consider the classic problem book coauthored by Lightman, Press, Price, and Teukolsky.

There are several other excellent textbooks http://www.math.ucr.edu/home/baez/RelWWW/reading.html#gtr (to that list, add Hartle, which appeared after I compiled it.)

In general, there is no reason to avoid the most widely used textbooks, but many good reasons to use one of them as your first book. Many of these should be available in inexpensive paperback editions via on-line booksellers.

Schaum's Outlines books are often quite good, but you really should not need to study such a book before plunging into gtr. Far more essential is that you have a solid grasp of the spacetime interpretation of str (see Spacetime Physics, by Taylor and Wheeler, FIRST edition). It seems especially unwise to subject yourself to studying a book on tensor calculus for its own sake if you truly "hate pure math".

 

[mathwonk]

i wouod recommend petes website on tensors (sear ch for threds on tensors), or ruslan sharipov's book . it seems to be pleasing to people who have tried it for computational facility with tensors. if you then also want conceptual treatments there are many possible sources in pure math books and general relativity books like wheeler, misner, thorne?

 

[coalquay404]

The answer to your question surely depends upon how seriously you want to study GR. If, as you say, you hate pure maths, you're probably going to be better served by reading a book which will give you a good qualitative introduction to the theory. At the most elementary level, Bob Geroch's General Relativity from A to B provides a nice non-mathematical account of the theory. At the next level I would probably choose to use Sean Carroll's book - it doesn't use a great deal of "difficult" mathematics and is certainly a far from rigorous treatment of the subject, but it does discuss a great deal of interesting material. It's certainly what I would choose to start with were I in your position.

As these threads seem to pop up so often, it might be worth pointing out a couple of books which would suit the other camp, i.e., those who aren't afraid of getting their hands dirty by considering mathematical rigour. In my opinion, MTW, while in many respects an interesting book, is an absolutely horrible text to learn GR from. It has a sort of chatty, informal style which I feel contributes little and often gets in the way of a clear discussion of the material. It is, however, a great book to dip into as bedtime reading once you're already competent in the subject (if your idea of fun bedtime reading involves manipulating a couple of kilograms of paper, that is). If you're interested in learning the mathematical aspects of GR, there are probably no books out there at the moment which deal with the mathematics in an acceptable way alongside the physics. My own advice would be to read, then re-read, then read again, the relevant chapters in Choquet-Bruhat/Dewitt-Morette/Dillard-Bleick's Analysis, Manifolds, and Physics, vol. I (the first author may or may not have been credited as Fouret-Bruhat, depending upon which edition you can lay your hands on). As a companion to this it's probably a good idea to have a more heuristic understanding of topology and Riemannian geometry to the level of, say, Nakahara's book. Only then would you be well-served by moving to Wald (Wald's book, by the way, contains a small but significant number of very serious mistakes, both conceptual and formal, which one needs to watch out for carefully. His appendix on Lagrangian and Hamiltonian field theory, in particular, contains serious errors and, if I recall correctly, his chapter on spinors needs a serious rewrite. Unfortunately none of these mistakes seem to be being corrected through reprints.) Apart from Hawking & Ellis, Wald should provide those of a mathematical bent with all they need to know about GR to start reading the literature and, as far as I'm concerned, there isn't a single other book out there which comes close to these two.

 

[ObsessiveMathsFreak]

You hate pure math!? How on Earth can you hate pure math but love (I guess) theoretical physics?

I imagine it would be pretty easy for many to come to such a conclusion.

Pure Mathematics:
A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S.

Not So Pure Mathematics:
A subset of a space is compact if it is closed.

You'd be surprised how many people are put off by the first explanation, even if it is more correct. The key point is that to understand the first explanation, you would have to have first grasped the second completely, along with its drawbacks.

 

[George Jones]

... and, if I recall correctly, his chapter on spinors needs a serious rewrite.

On page 350, Wald states

Similarly, the group ISL(2,C) - defined by composing in the natural way the elements of SL(2,C) with the elements of the two-complex-dimensional translation group of W - can be seen to be the universal covering group of the proper Poincare group.

In other words, Wald says that the universal covering group of the proper Poincare group is (isomorphic to) the semidirect product of SL(2,C) with C^2, when, actually, the universal covering group of the proper Poincare group is (isomorphic to) the semidirect product of SL(2,C) with R^4.

Greg Egan http://groups.google.ca/group/sci.p...f37233c400?lnk=st&q=&rnum=2#c5cd48f37233c400" a few years ago.

You might, however, have other, more egregious, examples in mind.

 

[mathwonk]

but the second statement in post 15 is false. is that why some people like it better, or is that allowed as a drawback?

 

[George Jones]

but the second statement in post 15 is false. is that why some people like it better, or is that allowed as a drawback?

A bit of elaboration on what mathwonk means.

I suspect that ObsessiveMathsFreak meant to state what is sometimes (e.g., blue Rudin) called the Heine-Borel theorem: a subset of R^n is compact iff it is closed and bounded.

As it stands, ObsessiveMathsFreak's second statement doesn't even work for R: [0, infinity) is closed but not compact.

 

[ObsessiveMathsFreak]

I suspect that ObsessiveMathsFreak meant to state what is sometimes (e.g., blue Rudin) called the Heine-Borel theorem: a subset of R^n is compact iff it is closed and bounded.

Yes, that's what I was getting at. For finite domains in R^n, compact means closed. I left out the bounded part.

Basically, you have to first know that compactness is an extension of closed boundedness. But without knowing about closed boundedness, I can't imagine that many people would appreciate compactness.

I imagine most people don't like pure mathematics much because it is a subject forever running before it walks. Introducing compactness without ever discussiing closed boundedness. Without referring to the latter, the former is relatively meaningless upon presentation. I myself encountered the pure definition of compactness and was mislead by it for a fairly substantial amount of time. I proved theorems with it without ever knowing what compactness actually was. The proofs were perfectly valid, but devoid of semantics for me.

Other less "pure" forms of mathematics would start at the beginning and might never get to compactness. But at least they would get somewhere someone can appreciate. Mathematics is about concepts as well as definitions. Definitions are just the way we present our concepts, not the concepts themselves. Pure mathematics often forgets that.

 

[Chris Hillman]

Huh?

I imagine it would be pretty easy for many to come to such a conclusion.

Pure Mathematics:
A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S.

Not So Pure Mathematics:
A subset of a space is compact if it is closed.

You'd be surprised how many people are put off by the first explanation, even if it is more correct.

We are wandering farther and farther off topic, I think, but I don't like to let that pass. Hmm... where to begin...

First, those aren't "explanations" but, in essence, definitions. As you know, mathematicians call an explanation of why some statement holds true a "proof"! And I guess we might call other types of explanations "context" or "discussion" or even "exposition".

Why do I say the statements quoted above are in essence definitions?
Well, the first definitions of "topology" and "compact" were not logically equivalent to the definitions which later became standard, and which really are preferable for many reasons (some general topology textbooks take the time to discuss why this is the case!). There are in fact many competing ways in which one could try to capture intuitive ideas of "nearby point", "continuous function", and so on, and the fathers of topology proved theorems relating some of these to each other. To some extent what concept you choose to call "topology on a set" or "continuous function" or "compact set" without qualification is a matter of convention, but as I say, the standard choices are well motivated. At this point, let me say that the best short introduction to general topology I know is Chapter 4 in the superb textbook by Gerald Folland, Real Analysis, Wiley, 1984.

Second, as you probably know, the Heine-Borel theorem states that in \bold{R}^p with the standard metric topology, a modification of your second proposed definition ("closed and bounded") is equivalent to the first. This is an example of what I said above, "the fathers of topology proved theorems relating some of these to each other". The added condition "bounded" is crucial since for example \bold{R}^p itself is closed, but not bounded, and it is not compact according any reasonable definition.

Third, there exist important topologies on important sets which do not behave anything like metric topologies.

 

[Chris Hillman]

Hi all,

I wrote my last reply before I realized that some of the points I made had already been addressed. Sorry!

Mathematics is about concepts as well as definitions. Definitions are just the way we present our concepts, not the concepts themselves.

Well, from this it seems that our viewpoint is probably pretty close after all.

Pure mathematics often forgets that.

I think that masters of analysis instinctively appreciate that competing definitions represent competing attempts to capture some intuition based upon insight into specific examples. And that which definitions are "best" is to some extent a matter of convention--- but more importantly, largely a matter of which are most convenient for stating and proving interesting and useful theorems. It follows that to some extent the latter judgement probably involves an unknown number of historical accidents, so is again somewhat arbitrary.

I think your real complaint (no pun intended) is that mathematics courses tend to be so crowded with material that there is often little time for the instructor to even mention the kind of important points we are discussing here. Besides the need to get through a crowded curriculum without rushing the "average students" in the class, another valid pedagogical consideration is that, even in the best schools, in an undergraduate real analysis course with an enrollment of say thirty students, only the five students with the most insight are likely to appreciate the sort of thing we are discussing here, so many instructors encourage those five to come to office hours to get more pointers on the "why".

Some weeks ago, I suggested in another thread that I feel that the world would benefit greatly if the "standard time" in which one can expect to get through college with a math/sci/engineering major were to be extended from four to six years. This would give designers of curricula more leeway to "open up" courses which are neccessarily crowded with crucial concepts and techniques to include explicit discussion of what material represents logical neccessity and what incorporates some degree of convention or (often rather intelligent) choice, and various other kinds of "why" questions. IOW, to include more "context" or even, dare I say it, philosophy.

There is another school of thought which maintains that if anything the standard time should be shortened. Indeed, a valid objection to my proposal is that many students are simply to impatient to sit through four years of college, much less six.

This brings us to another huge issue on teaching math/sci courses, which many students don't fully appreciate until they start teaching themselves (if this happens). Namely: students exhibit huge and often unpredictable individual variations in how they think (or can be readily taught to think), and thus how they learn (or rather, how they learn most efficiently and with the least amount of uneccessary suffering). There is really no simple answer to this problem. All teachers struggle with it constantly, which means that all students are also constantly being affected by whatever decisions the instructor makes "in real time" about how to answer a query in class (e.g., when to say "see me in office hours").

 

[Chris Hillman]

Mistakes in widely cited textbooks

(Wald's book, by the way, contains a small but significant number of very serious mistakes, both conceptual and formal, which one needs to watch out for carefully. His appendix on Lagrangian and Hamiltonian field theory, in particular, contains serious errors and, if I recall correctly, his chapter on spinors needs a serious rewrite. Unfortunately none of these mistakes seem to be being corrected through reprints.)

This point is too important to be left inside those parentheses, I think!

I am not sure that I know of any textbook on any subject which is entirely free of any infelicities of exposition, or even of any trace of outright error.

(BTW, I think that MTW comes surprisingly close to being virtually error free, although as we can see from comments in this thread, commentators seem to either love this book or to fear it.)

If first year graduate students didn't have to study for quals, some summer a band of same would no doubt compile a list of errata for the dozen most widely used gtr textbooks, and put them up somewhere for the benefit of all (naturally not omitting to email the appropriate authors).

 

[George Jones]

Yes, that's what I was getting at. For finite domains in R^n, compact means closed. I left out the bounded part.

Basically, you have to first know that compactness is an extension of closed boundedness. But without knowing about closed boundedness, I can't imagine that many people would appreciate compactness.

This might be true. I took two semesters of second-year real analysis in which compactness was first covered in R^n in terms of the Heine-Borel theorem and sequential compactness. Only after this (near the end of the second semester) did we take the general definition of compactness.

Later, I took a topology course that treated compactness in terms of nets. The concept of a net is a generalization of the concept of a sequence.

A subset A of a topological is not compact if A is "so big" that there exists a least one net in A that does not have an accumulation point, i.e., for every point p in A, there is a neighbourhood of p that the net eventually stays out of.

 

[mathwonk]

to me some posters are missing the point of compactness. it is not a generalization of closed boudnedness, but a generalization of finiteness.

i.e. most anything you can prove for finite sets you can also prove for compact ones, in the presence of continuity.

so the heine borel definition actually states rather clearly the key property that compact sets have and that is useful.

it is then an important goal to find some compact sets. the theorem that in R^n closed bounded ones have this key compactness property is such a result.

so I guess you need both results. i.e. closed boundedness tells you nothing but what they are good for, while the heine borel property tells you nothing about how to recognize one (the defining property being uncheckable directly).

 

[mathwonk]

actually this is not far off topic at all. mathematics is about identifying the key prepoerties present in a given situation and making them stand out by appropriate definitions.

the tensor books explaining primarily tensor calculations are like the books that say only that compactness means closed and bounded. I.e. some do not explain what is going on.

I advocate learning also what they really are, i.e. how tensors behave, and what they mean, and what their key proeprties are, like multilinearity.

It is useful to know all aspects, defining properties, basic manipulations, and recognition theorems. i.e. where do tensors come from and where are they found? what are their key properties? how do i compute with them?

tensors have two aspects, local and global. locally they are a generalization of polynomials, only they can be noncommutative. to represent them by numbers, one uses coordinates, whence they, like polynomials, are represented by their coefficients. then one needs to know how the coefficients change when the local coordinates are changed.

where do they come from? mathematiocally they arise from trying to multiply tangent vectors and cotangent vectors. physically? the physicists can say better than i. presumably from trying to calculate certain mechanical phenomena like stress.

or to bring us back home, perhaps when einstein used them to link gravitational forces to geometric curvature, since tensors are the right tool to express curvature. if so, i conjecture a simple study of curved surfaces in clasical differential geometry "done right" e.g. from theodore shifrin's notes, would not hurt a budding relativist.

 

[Thrice]

Hi. I was wondering if you guys can refer me to the best possible text that teaches the computations in tensor analysis. I hate pure math. I just want to get a feel for the mathematics before engaging the physics.

As far as I know this is a widely referred text, however, it doesn't come with a solution manual.

https://www.amazon.com/gp/product/0070334846/?tag=pfamazon01-20

Thanks

1) use multiple books
2) browse every book before you buy it. Use libraries if there are any near you.
3) don't forget the free books.

 

[Doodle Bob]

1) use multiple books
2) browse every book before you buy it. Use libraries if there are any near you.
3) don't forget the free books.

Better yet: take a class in the subject!

 

[mathwonk]

if i have the energy and time, it might be fun to start a threqd reading MTW here.

 

[George Jones]

Thinking about some stuff in another thread reminded me of an application of compactness in relativity. The open cover definition of compactness can be used to give a nice little proof that every compact spacetime manifold admits closed timelike curves (time travel).

 

[mathwonk]

Good example,

Indeed, that is the whole point of the heine borel formulation of compactness, it is easy to deduce other important things from it.

i.e. the most important thing you cAN SAY about closed bounded sets, is that they have the heine borel property. then you can deduce myriad other things.i.e. when one considers clsoed bounded sets, one should ask what is their most useful feature. eventually one comws up with the fact that for cliosed boudned sets, every open cover has a finte subcover.

THEN, seing this as the key proeprty, one changes this theorem into a definition, i.e. anything sharing this property that clsoed bounded sets possess, is called compact.

in this sense compact sets are an abstraction of closed bounded sets, which share only their key feature.

we do this in mathematics all the time. we study the integers, and sue the division algorithm to prove unique factorization. seing this, we then define a "Euclidean domain" to be any domain with a division algorithm. then we easily deduce that every euclidean domain has unique factorization.

the definition becomes useful when we encounter new situations where the euclidean again holds, like for polynomials over a field.

similarly we encounter new spaces which have the heine borel property , such as any metric space which is complete and totally bounded. we sefor example that,mperhaps surprizingly, the unit ball in hilbert space of infinite dimensions is not compact, although closed and bounded.

'then we begin to see that clsoed and bounded is not the essential proeprty for compactness.

 

[Daverz]

if i have the energy and time, it might be fun to start a threqd reading MTW here.

Having too much time on my hands, I'd be willing to help "proctor" such a thead (which should probably be in the SR&GR forum).

 

[chroot]

I'd be willing to participate in an MTW thread, also.

- Warren